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Uniformity

For certain reasons, this is something of a follow-up to an earlier article, Interconnectness, in which I quoted myself from December of 1987 as saying,

I came to think that if one understood the law of contradiction, there would be nothing left to understand.

I am going to quote now the person I quote the most, Collingwood, from the fourth paragraph from the end of Religion and Philosophy:

Uniformity, in a word, is relative to our needs; and to suggest that a game of cricket, for instance, would be impossible if we supposed that the ball might suddenly decide to fly to the moon, is no less and no more sensible than to suggest that it is impossible because the bowler might put it in his pocket and walk off the field. We know that the friend we trust is abstractly capable, if he wished, of betraying us, but that does not prevent our trusting him. It may be that our faith in the uniformity of matter is less removed from such a trust than we sometimes imagine. Whether we describe it as faith in matter or faith in God makes, after all we have said, little difference.

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Hands on ≠ Minds on

I “reblog” an article about a history lesson in which students knock down a mock-up of the Berlin Wall. I think this “reenactment” (in the quotation in the re-blogged post) of the demolition of the Berlin Wall is just what Collingwood said (in An Autobiography) was not doing history. Here follows Collingwood:

I expressed this new conception of history in the phrase: `all history is the history of thought.’ You are thinking historically, I meant, when you say about anything, `I see what the person who made this (wrote this, used this, designed this, &c.) was thinking.’ until you can say that, you may be trying to think historically, but you are not succeeding. And there is nothing except thought that can be the object of historical knowledge. Political history is the history of political thought: not `political theory’, but the thought which occupies the mind of a man engaged in political work: the formation of a policy, the planning of means to execute it, the attempt to carry it into effect, the discovery that others are hostile to it, the devising of ways to overcome their hostility, and so forth…Military history, again, is not a description of weary marches in heat or cold, or the thrills and chills of battle or the long agony of wounded men. It is a description of plans and counter-plans: of thinking about strategy and thinking about tactics, and in the last resort of what men in the ranks thought about the battle.

On what conditions was it possible to know the history of a thought? First, the thought must be expressed: either in what we call language, or in one of the many other forms of expressive activity…Secondly, the historian must be able to think over again for himself the thought whose expression he is trying to interpret…If some one, hereinafter called the mathematician, has written that twice two is four, and if some one else, hereinafter called the historian, wants to know what he was thinking when he made those marks on paper, the historian will never be able to answer this question unless he is mathematician enough to think exactly what the mathematician thought, and expressed by writing that twice two are four. When he interprets the marks on paper, and says, `by these marks the mathematician meant that twice two are four’, he is thinking simultaneously: (a) that twice two are four, (b) that the mathematician thought this, too; and (c) that he expressed this thought by making these marks on paper…

This gave me a second proposition: `historical knowledge is the re-enactment in the historian’s mind of the thought whose history he is studying.’

Granted, and...

from today’s Smartbrief:

Student members of the Young Americans for Freedom at a school in Rome, Ga., marked the 25th anniversary of the fall of the Berlin Wall and the reunification of Germany with a re-enactment at their school. They knocked down a graffiti-covered, 12-foot-long wall made from wood for the dramatization. “It is great to see them internalizing the lessons of history and exhibiting the power of freedom,” said Brad Poston, history department chair.

By that argument, burning down the school would be a rich learning activity in support of “internalizing the lessons of history” of the urban riots of the 60s.

When, oh when, will teachers truly understand the difference between fun activity and experiential well-designed learning?

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The Parabola

I do not now recall my specific inspiration; but in January of 2012, sitting at home in Istanbul, I cut up a cardboard box in order to make a model of a parabola quâ conic section.

January 14, 2012

January 14, 2012


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Two women

On the left, Vincent van Gogh, Mousmé, 1888, National Gallery of Art, Washington. On the right, Stephen Chambers, Woman (Green Background), 2006, private collection, London; currently on display at the Pera Museum, Istanbul, where I saw it on Saturday, June 21, 2014, and made the (slightly tilted) photograph on the right below.

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The van Gogh image, I downloaded from the National Gallery website. I cropped and resized the Chambers image to be the same height as the van Gogh; then I juxtaposed the two with convert +append van-gogh-la-mousme.jpg chambers-woman.jpg two-women.jpg

I did not know of the artist Stephen Chambers before. The green background of his Woman, and her expression, caused me to think of van Gogh’s Mousmé. The Mousmé’s head is round, but her dotted skirt is as flat as the blouse of the Woman.

Burgazada

Pressure

Istanbul is a crowded, paved city. Consider the graphic below, showing public green space in Istanbul, London, New York, Berlin, Hong Kong, and Paris. The green space of Istanbul is almost invisible.

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Istanbul green space is invisible in the graphic that inspired the one above:

Image from Twitter, @GeziParkii

Image from Twitter, @GeziParkii

I saw this last graphic on Twitter, but the data used to create it can be found in the table on pages 44–5 of World Cities Culture Report 2013. In fact the disk for each city seems to be drawn so that the diameter of the green disk is the given percentage of the diameter of the whole. This makes the green disk too small. The square root of the percentage should have been taken first, since circles are to one another as the squares on their diameters: Euclid shows us this in one of the most remarkable theorems of antiquity, Proposition 2 of Book XII of the Elements. I used this result to create the first graphic above. Istanbul still has not got a lot of green.

I have spent at least a few days in each of the cities depicted except Hong Kong. Not counting Istanbul, I suppose Paris is the most beautiful, although it has less parkland than the others. However, Istanbul would not be more beautiful with an Ottoman-style building in place of Gezi Parkı. The city has already lost open space that would have been wanted in the next earthquake.

Although the Mecidiyeköy neighborhood of Istanbul where we live was settled in the nineteenth century, most of the development there is only a few decades old. In the midst of this development, no land was set aside for parks. There is a large stand of trees near us that we see every day: it is a cemetery. Actually it is three adjacent cemeteries, for Greeks, Armenians, and Jews respectively. The cemeteries lie behind a high wall.

Istanbul’s lack of trees and open space is oppressive. Sidewalks are not wide enough. Drivers may say that there are not enough roads; they should be saying that there are too many cars, too few busses, and too few subway lines. More subway lines are being laid. Our university building is surrounded by skyscrapers under construction. One may smile on this development as one might smile on the smoke pouring from a factory chimney: it means productivity and jobs. It also means damage to health, both physical or mental.

When we lived in Ankara, our university was in the midst of a forest, which had been planted by the students when the university was built in the 1960s. This forest was a spiritual benefit, or a mitigation, of life in that also-crowded city. Unfortunately, since we left, the university forest has been invaded by the highway-building crews of the mayor.

Release

Last weekend we visited a part of Istanbul that cannot be so easily opened up to development. However, nothing is certain in that regard. Ayşe and I took a ferry to the Islands, specifically Burgazada. We stayed there the night of Saturday, February 8, 2014, just before the first week of classes of our spring semester.

Back on the European mainland, not far from our university and our home, police were attacking protesters of a new internet censorship law (passed by the Turkish parliament, not yet signed by the President of the Republic). Even under existing law, all WordPress blogs were blocked in Turkey a few years ago. (The present WordPress blog did not exist at the time.)

A foreign journalist called Mahir Zeynalov was deported from Turkey on Friday, February 7, ostensibly for tweeting against the state. I have retweeted some of Mr Zeynalov’s messages about the matter, if they contain links to news stories; but I have fewer than a hundred followers.

It was a delight to find in Burgazada an island of peace, literally, within a noisy crowded city. Maybe I should liken the island to the strawberry in a parable attributed to the Buddha:

A man traveling across a field encountered a tiger. He fled, the tiger after him. Coming to a precipice, he caught hold of the root of a wild vine and swung himself down over the edge. The tiger sniffed at him from above. Trembling, the man looked down to where, far below, another tiger was waiting to eat him. Only the vine sustained him.

Two mice, one white and one black, little by little started to gnaw away the vine. The man saw a luscious strawberry near him. Grasping the vine with one hand, he plucked the strawberry with the other. How sweet it tasted!

Etymology and history

In Byzantine times, the islands were where you were sent into exile after being blinded. In Turkish times, it seems they have been a refuge for Christians and Jews. The first mosque on Burgazada was built only in 1953—“in commemoration of the 500 year anniversary of the conquest of Istanbul in 1453” as the Istanbul municipal website triumphantly points out. The same website tells us helpfully about the economics of the island: “In the 1950s…a number of Jewish merchants settled in Burgazada. This caused a sharp increase in the price of housing. The very wealthy people who settled there built summer villas and houses along the hillsides above Heybeliada.” My guess is that wealthy Muslims wanting summer homes are now helping to drive out the native population. Year-round residence is declining.

The Greek name of Burgazada is Αντιγόνη, after Antigonus I Monophthalmus, one of Alexander’s satraps. Supposedly Antigonus’s son Demetrius built a tower there, but no trace remains—except in the name Burgaz, which seems to be derived from the Greek πύργος “tower”, although the Greek Wikipedia article does not give this word.

The iconodule Methodius was supposedly imprisoned in a dungeon on Burgazada. We visited the Church of St John built over the dungeon. At least we visited the narthex; the nave was locked. But first we saw Sait Faik’s house nearby.

What we did

Saturday started out cloudy, but it cleared up. From home, we caught a 7:40 bus to the ferry terminal at Kabataş. On the ferry we sat on deck, though eventually Ayşe moved into the cabin for warmth. I could have moved as well, to be away from cigarette smoke. Smoking is forbidden everywhere on ferries, but this just means that lighting up, at least on deck, is one form of civil disobedience that people engage in without hesitation. Sometimes I ask people not to smoke, but I do not like to do it.

On the island, we walked to the öğretmenevi or teachers’ lodge. On the island, you either walk or take a fayton. There are no private automobiles. The police use cars, and we saw a van belonging to the gas company. Trash is collected in a truck, and this makes no sense to me, since the trash that people produce is no more than what they have brought in by hand in the first place. Food garbage could be composted; empty packaging could be taken back to the shops by the ferry pier.

In the summer of 1999, Ayşe and I visited Burgazada, but the day was cool and rainy. We tried to visit the house of Sait Faik then, but it was closed. This time it was open. Various panels told (usually in Turkish only) the story of the writer.

Books from Sait Faik's library, here Baudelaire's Fleurs du Mal with a risque cover

Books from Sait Faik’s library, here Baudelaire’s Fleurs du Mal with a risque cover

Maugham's “Rain” and Hemingway's For Whom the Bell Tolls in Turkish translation

Maugham’s “Rain” and Hemingway’s For Whom the Bell Tolls in Turkish translation

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A view of the Church of St John

A view of the Church of St John

Sait Faik's membership in the International Mark Twain Society

Sait Faik’s membership in the International Mark Twain Society

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After lunch, we wandered about, eventually making our way to the top of the island’s one hill. Most of the island’s forest cover burned down in 2003, but trees at the peak somehow remained. In fact there are no direct signs of a fire. Everywhere is green. Planted trees are growing.

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As we walked up to the peak, two boys alternately rode and led two horses up there.

Near the peak was a Greek church and cemetery. A man was building a wall around the cemetery; a boy unlocked the church for us. The boy said the island’s horses were allowed the graze freely.

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I transcribe the text on an inside wall of the church. I mean I type it up; I am not translating. The oddities of the English are in the original. No other language was offered:

Christos Metamorphosis Monastery was built between the years 867 and 886, with the order of Byzantine King Basil I, the Macedonian. the name of the Monastery “Metamorphosis” refers to Jesus’ metamorphosis description; hence the monastery was dedicated to Jesus.

The monastery is built on top of the remaining of Ancient Greek God Zeus’s sanctuary. According to a story, the Monastery was ruined by people since the fire lit during the services was perceived as fire by public and caused chaos. Remaining of the ruined Monastery is then used in the construction of Aghios Ioannes (Aya Yani) Church and Agios Georgios (Aya Yorgi) Karipi Monastery, again located in Burgazada. After a while, a professor in Halki Academy of Commerce, Hurmuzis Triantafilu built a relatively smaller Church (the Church we see today), with great efforts, by collecting donations. In June 22, 1869, the Church started the service. When Triantafilu died in October 6, 1882 he was buried to the cemetery that is located next to the Church. In year 1928 the Monastery’s management is given to the Patriarchate Monastery’s Commission. Next to the Monastery there is the Burgazada Greek Cemetery. The building of the old Monastery is being used as the lodging for the cemetery’s keeper. Every year August 6th is celebrated as the Monastery’s day.

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Some of the graves in the cemetery were covered by stone slabs, others by dirt alone in what I understand to be the Muslim style.

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A use of the camera's panorama feature

A use of the camera’s panorama feature

The Asian-side development does not look bad from a distance

The Asian-side development does not look bad from a distance

The view of Heybeliada through some of Burgazada's unburnt trees

The view of Heybeliada through some of Burgazada’s unburnt trees

Sunset from the balcony of the suite at the teachers' lodge

Sunset from the balcony of the suite at the teachers’ lodge

From the balcony again, Sunday morning

From the balcony again, Sunday morning

A third Greek church on the island

A third Greek church on the island

Wandering on Sunday morning, we found the working class district, or at least the stables where the horses that draw the phaetons are kept at night. The smell of manure pervades the area. The drivers seem to live in the same place.

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I can appreciate the sentiment, at least, of a graffitist. But telling people not to use phaetons will be as effective as telling them not to smoke. And what should the drivers of phaetons do for their livelihoods? I would suggest they be given pedicabs.

“The phaeton is a cruelty to horses”

“The phaeton is a cruelty to horses”

“Don't ride a phaeton, ride a bike!”

“Don’t ride a phaeton, ride a bike!”

Yassıada (“flat island”) and Sivriada (“pointed island”) were visible to the west. The former is occupied by the military.

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We came to the restaurant at Kalpazankaya, “counterfeiter rock.” This was the end of the coastal road. But there were stairs down to the shore.

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There was a lovely picnic spot, remarkably clean.

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I tried to continue the counterclockwise tour of the island, but eventually the cliffs made it impossible.

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Back at the pier, we had a Turkish coffee while waiting for the ferry.

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Give childhood back to children

Give childhood back to children.

via Give childhood back to children.

I created this article by pressing a button beneath the friend’s article that is linked to above. That article links in turn to an article by Peter Gray in The Independent with the headline “Give childhood back to children: if we want our offspring to have happy, productive and moral lives, we must allow more time for play, not less”. Gray writes,

I’m lucky. I grew up in the United States in the 1950s, at the tail end of what the historian Howard Chudacoff refers to as the “golden age” of children’s free play. The need for child labour had declined greatly, decades earlier, and adults had not yet begun to take away the freedom that children had gained.

I don’t think Gray quite says this, but it seems to me that making young people study in school for the sake of their future remunerative employment is just another form of child labor, even if it is supposed to be for their own good. As angry children are supposedly wont to say, they didn’t ask to be born in the first place.

Learning mathematics

This is mostly reminiscences about high school. I also give some opinions about how mathematics ought to be learned. This article originally formed one piece with my last article, “Limits”.

I learned calculus, and the epsilon-delta definition of limit, in Washington D.C., in the last two years of high school, in a course taught by a peculiar fellow named Donald J. Brown. The first of these two years was officially called Precalculus Honors, but some time in that year, we started in on calculus proper. Continue reading

Limits

This is about limits in mathematics: both the technical notion that arises in calculus, and the barriers to comprehension that one might reach in one’s own studies. I am going to say a few technical things about the technical notion, but there is no reason why this should be a barrier to your reading: you can just skip the paragraphs that have special symbols in them.

Looking up something else in the online magazine called Slate, I noted a reprint of an article called “What It Feels Like to Be Bad at Math” from a blog called Math With Bad Drawings by Ben Orlin. Now teaching high-school mathematics, Mr Orlin recalls his difficulties in an undergraduate topology course. His memories help him understand the difficulties of his own students. When students do not study, why is this? It is because studying makes them conscious of how much they do not understand. They feel stupid, and they do not like this feeling. Continue reading

Self-similarity

Animation: circles within circles

From the poster depicting a few von Neumann natural numbers, I created this animation. The moving image no longer depicts natural numbers in the sense of the poster, since there is no infinite descending chain of natural numbers. There is an infinite ascending chain of them; but the poster does not actually depict such a chain as nested circles. So running the animation in reverse would not give a correct suggestion of the original poster, even if it were of infinite size. Continue reading

Hello world!

When I learn things that might be worth remembering, and when I have thoughts that might be worth pursuing, it may be useful or convenient to type them up as here. I begin with the opening paragraph of the third of the 77 short chapters of W. Somerset Maugham’s book called The Summing Up (published in 1938 when Maugham was about 64): Continue reading